Graeme's comment: very interesting geometrical things, especially involving generalized circles and inversion geometry, and some helpful cut-the-knot-like applets.  Well worth a visit.

and other mathematical constants:

http://oldweb.cecm.sfu.ca/pi/pi.html 

http://www.cecm.sfu.ca/~pborwein/PISTUFF/Apistuff.html

http://www.mathsoft.com/mathsoft_resources/mathsoft_constants/ lists a bunch of constants, but even more are listed by

http://pauillac.inria.fr/algo/bsolve/constant/constant.html , which lists these constants:

Well-known constants

Zero, 0

One, 1

Imaginary unit,

Pythagoras' constant,

Golden mean,

Natural logarithmic base, e

Archimedes' constant,

Euler-Mascheroni constant,

Apéry's constant,

Catalan's constant, G

Khintchine's constant

Feigenbaum constants

Madelung's constant

Chaitin's constant

Constants associated with Number Theory

Hardy-Littlewood constants

Meissel-Mertens constants

Euler gamma function constants

Landau-Ramanujan constant

Sierpinski's constant

Brun's constant

Artin's constant

Linnik's constant

Hafner-Sarnak-McCurley constant

Gauss-Kuzmin-Wirsing constant

Stolarsky-Harborth constant

Porter's constant

Glaisher-Kinkelin constant

Fransén-Robinson constant

Alladi-Grinstead constant

Niven's constant

Backhouse's constant

Mills' constant

Stieltjes constants

Liouville-Roth constants

Diophantine approximation constants

Erdös' reciprocal sum constants

Abundant numbers density constant

Self-numbers density constant

Ramanujan-Soldner constant

Cameron's sum-free set constants

Euler totient function asymptotic constants

Pisot-Vijayaraghavan-Salem constants

Nielsen-Ramanujan constants

Triple-free set constants

De Bruijn-Newman constant

Komornik-Loreti constant

Freiman's constant

Erdös-Lebensold constant

Cahen's constant

Hall-Montgomery constant

Fast matrix multiplication constants

Erdös' sum-distinct set constant

Vallée's constant

Pell-Stevenhagen constants

Constants associated with Analytic Inequalities

Shapiro's cyclic sum constant

Carlson-Levin constants

Landau-Kolmogorov constants

Hilbert's constants

Copson-de Bruijn constants

Wirtinger-Sobolev isoperimetric constants

Whitney-Mikhlin extension constants

Korn constants

Kneser-Mahler polynomial constants

Grothendieck's constants

Du Bois Reymond's constants

Steinitz constants

Young-Fejér-Jackson constants

Van der Corput's constant

Constants associated with the Approximation of Functions

Wilbraham-Gibbs constant

Lebesgue constants

Favard constants

Bernstein's constant

The "one-ninth" constant

Laplace limit constant

Integer Chebyshev constant

Constants associated with Enumerating Discrete Structures

Abelian group enumeration constants

Pythagorean triple constants

Rényi's parking constants

Golomb-Dickman constant

Lengyel's constant

Otter's tree enumeration constants

Pólya's random walk constants

Self-avoiding-walk connective constants

Feller's coin tossing constants

Hard square entropy constant

Binary search tree constants

Digital search tree constants

Quadtree constants

Extreme value constants

Pattern-free word constants

Takeuchi-Prellberg constant

Percolation cluster density constants

Klarner's lattice animal constant

Lenz-Ising constants

2D Monomer-dimer constant

3D dimer constant

Lieb's square ice constant

Odlyzko-Rains constants

Tutte-Beraha constants

Constants associated with Functional Iteration

Gauss' lemniscate constant

Grossman's constant

Plouffe's constant

Lehmer's constant

Iterated exponential constants

Continued fraction constants

Infinite product constants

Quadratic recurrence constants

Conway's constant

Constants associated with Complex Analysis

Bloch-Landau constants

Masser-Gramain constant

John constant

Hayman constants

Littlewood-Clunie-Pommerenke constants

Whittaker-Goncarov constants

Grötzsch ring constants

Constants associated with Geometry

Geometric probability constants

Circular coverage constants

Universal coverage constants

Hermite's constants

Tammes' constants

Calabi's triangle constant

DeVicci's tesseract constant

Graham's hexagon constant

Traveling salesman constants

Steiner tree constants

Moving sofa constant

Beam detection constant

Heilbronn triangle constants

Kakeya-Besicovitch constants

Moser's worm constant

Rectilinear crossing constant

Maximum inradius constants

Apollonian packing constant

Magic geometric constants

Relevant Links

Inverse Symbolic Calculator and Integer Relations (CECM)

Numbers, Constants and Computation, by X. Gourdon and P. Sebah

Recognizing Numerical Constants, by D. H. Bailey and S. Plouffe

What is a Number?, by A. Bogomolny

What is a Closed-Form Number?, by T. Y. Chow

How Many Fundamental Physical Constants are There?, by J. Baez

On the Random Character of Fundamental Constant Expansions, by D. H. Bailey and R. E. Crandall

University of Tennesse Mathematics Archives Page about Numbers

Math Forum Internet Mathematics Library Page about Numbers

Power Sums of Complex Numbers

Polynomial Optimization Over an Ellipse

Yahoo Page about Numbers